Preheat grill to medium heat.
In a blender, combine the garlic, lemon juice and Dijon mustard. Drizzle in 3/4 cup olive oil to emulsify. Add the Parmesan and pulse.
Gently toss the hearts of romaine in remaining olive oil and season with salt and pepper. Grill for 2 minutes, until grill marks appear and the romaine becomes wilted.
Remove the lettuce to a cutting board and cut into 2-inch wide strips. Place in a bowl and drizzle with the dressing. Toss to coat and serve warm. Garnish with the shaved Parmesan.